{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:LV22X6ZNTHRCSBZQE2YVMVAS4P","short_pith_number":"pith:LV22X6ZN","canonical_record":{"source":{"id":"2510.23369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-10-27T14:20:01Z","cross_cats_sorted":["math.RA","math.RT"],"title_canon_sha256":"57f1df92f029739eefe31eea65ebc53a2cdb42e2d93de6266b3d902b4fb04adc","abstract_canon_sha256":"9cbf9c7df81c4c8094cdd633f1cbc61bce915bf34ef3a947c1138adf443fe7b7"},"schema_version":"1.0"},"canonical_sha256":"5d75abfb2d99e229073026b1565412e3fafb07b89cfc85171239ce2089a9e883","source":{"kind":"arxiv","id":"2510.23369","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.23369","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"arxiv_version","alias_value":"2510.23369v2","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.23369","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_12","alias_value":"LV22X6ZNTHRC","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_16","alias_value":"LV22X6ZNTHRCSBZQ","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_8","alias_value":"LV22X6ZN","created_at":"2026-05-25T02:01:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:LV22X6ZNTHRCSBZQE2YVMVAS4P","target":"record","payload":{"canonical_record":{"source":{"id":"2510.23369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-10-27T14:20:01Z","cross_cats_sorted":["math.RA","math.RT"],"title_canon_sha256":"57f1df92f029739eefe31eea65ebc53a2cdb42e2d93de6266b3d902b4fb04adc","abstract_canon_sha256":"9cbf9c7df81c4c8094cdd633f1cbc61bce915bf34ef3a947c1138adf443fe7b7"},"schema_version":"1.0"},"canonical_sha256":"5d75abfb2d99e229073026b1565412e3fafb07b89cfc85171239ce2089a9e883","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:08.919028Z","signature_b64":"fU+d6sm21jVfRXWh0HxtgnsDvsdiigpG6cA/BiS89ZyAvnvOepg3Ltr1OPaCqmoKqOoZQaQ8B2mRerU+aNBGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d75abfb2d99e229073026b1565412e3fafb07b89cfc85171239ce2089a9e883","last_reissued_at":"2026-05-25T02:01:08.918222Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:08.918222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2510.23369","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:01:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g474G1KfeiI9garz66qIeCuRqpD4FXaj5gdbrD7GpX/kmAcWqaASWuuCr75fNS0jfou6Hn1dF9BHXFj3k+rGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:28:56.671447Z"},"content_sha256":"8c5516f8661b4ea833e5c4ee98dd7126252869383a7d50f42fc55367a323ecfb","schema_version":"1.0","event_id":"sha256:8c5516f8661b4ea833e5c4ee98dd7126252869383a7d50f42fc55367a323ecfb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:LV22X6ZNTHRCSBZQE2YVMVAS4P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the equivalence between the existence of $n$-kernels and $n$-cokernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","cross_cats":["math.RA","math.RT"],"primary_cat":"math.CT","authors_text":"Vitor Gulisz, Wolfgang Rump","submitted_at":"2025-10-27T14:20:01Z","abstract_excerpt":"We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ac0de6feac25256130574af315cf3be3267e92e54e532c47861d3e4df627fe66"},"source":{"id":"2510.23369","kind":"arxiv","version":2},"verdict":{"id":"54be0160-6d03-4ab7-8863-f65b306216a7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T03:50:37.119395Z","strongest_claim":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer.","one_line_summary":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem).","pith_extraction_headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.23369/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"f86edd0fdd926385945914e3d7f0cb267d8301f6461086caeb9b8e667709b3b9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"54be0160-6d03-4ab7-8863-f65b306216a7"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:01:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aJeVaoNMBYKLWji1IxkG4QM5hhQ7hfnWjS67bF8Ly28RY8HVWlwmQJqPbd1qM8wpM1ibNCo0LibuJKFy7mY0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:28:56.672149Z"},"content_sha256":"9ecde1a55dad6c65c4c4544c7c06ad21d4afd6f4bd6c780b3fb287879151bce3","schema_version":"1.0","event_id":"sha256:9ecde1a55dad6c65c4c4544c7c06ad21d4afd6f4bd6c780b3fb287879151bce3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/bundle.json","state_url":"https://pith.science/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T21:28:56Z","links":{"resolver":"https://pith.science/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P","bundle":"https://pith.science/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/bundle.json","state":"https://pith.science/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LV22X6ZNTHRCSBZQE2YVMVAS4P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:LV22X6ZNTHRCSBZQE2YVMVAS4P","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9cbf9c7df81c4c8094cdd633f1cbc61bce915bf34ef3a947c1138adf443fe7b7","cross_cats_sorted":["math.RA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-10-27T14:20:01Z","title_canon_sha256":"57f1df92f029739eefe31eea65ebc53a2cdb42e2d93de6266b3d902b4fb04adc"},"schema_version":"1.0","source":{"id":"2510.23369","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.23369","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"arxiv_version","alias_value":"2510.23369v2","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.23369","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_12","alias_value":"LV22X6ZNTHRC","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_16","alias_value":"LV22X6ZNTHRCSBZQ","created_at":"2026-05-25T02:01:08Z"},{"alias_kind":"pith_short_8","alias_value":"LV22X6ZN","created_at":"2026-05-25T02:01:08Z"}],"graph_snapshots":[{"event_id":"sha256:9ecde1a55dad6c65c4c4544c7c06ad21d4afd6f4bd6c780b3fb287879151bce3","target":"graph","created_at":"2026-05-25T02:01:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem)."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do."}],"snapshot_sha256":"ac0de6feac25256130574af315cf3be3267e92e54e532c47861d3e4df627fe66"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"f86edd0fdd926385945914e3d7f0cb267d8301f6461086caeb9b8e667709b3b9"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2510.23369/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained.","authors_text":"Vitor Gulisz, Wolfgang Rump","cross_cats":["math.RA","math.RT"],"headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-10-27T14:20:01Z","title":"On the equivalence between the existence of $n$-kernels and $n$-cokernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.23369","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-18T03:50:37.119395Z","id":"54be0160-6d03-4ab7-8863-f65b306216a7","model_set":{"reader":"grok-4.3"},"one_line_summary":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","strongest_claim":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer.","weakest_assumption":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem)."}},"verdict_id":"54be0160-6d03-4ab7-8863-f65b306216a7"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8c5516f8661b4ea833e5c4ee98dd7126252869383a7d50f42fc55367a323ecfb","target":"record","created_at":"2026-05-25T02:01:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9cbf9c7df81c4c8094cdd633f1cbc61bce915bf34ef3a947c1138adf443fe7b7","cross_cats_sorted":["math.RA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-10-27T14:20:01Z","title_canon_sha256":"57f1df92f029739eefe31eea65ebc53a2cdb42e2d93de6266b3d902b4fb04adc"},"schema_version":"1.0","source":{"id":"2510.23369","kind":"arxiv","version":2}},"canonical_sha256":"5d75abfb2d99e229073026b1565412e3fafb07b89cfc85171239ce2089a9e883","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5d75abfb2d99e229073026b1565412e3fafb07b89cfc85171239ce2089a9e883","first_computed_at":"2026-05-25T02:01:08.918222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:01:08.918222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fU+d6sm21jVfRXWh0HxtgnsDvsdiigpG6cA/BiS89ZyAvnvOepg3Ltr1OPaCqmoKqOoZQaQ8B2mRerU+aNBGDQ==","signature_status":"signed_v1","signed_at":"2026-05-25T02:01:08.919028Z","signed_message":"canonical_sha256_bytes"},"source_id":"2510.23369","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c5516f8661b4ea833e5c4ee98dd7126252869383a7d50f42fc55367a323ecfb","sha256:9ecde1a55dad6c65c4c4544c7c06ad21d4afd6f4bd6c780b3fb287879151bce3"],"state_sha256":"668b414ef7ebb9a551c1fbbec2d4fa39c86870716a8637aa07e4ac9d79ab0a27"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f9lWJzjix69N8hZLxgJDY5kke9NI2ewv9eKCq6q7WdMaNoK15bPS4wwGp0pwTlAKsm+RHaKoro2CVUySGwV8AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T21:28:56.676100Z","bundle_sha256":"00b78018b35a31b9714ef865bd271bc042b73ef3e6c5718da91dff2de2366193"}}